docs: def two_phase_dP_acceleration(m, D, xi, xo, alpha_i, alpha_o, rho_li, rho_gi,  
                                rho_lo=None, rho_go=None):
      r'''This function handles calculation of two-phase liquid-gas pressure drop
      due to acceleration for flow inside channels. This is a discrete 
      calculation for a segment with a known difference in quality (and ideally
      known inlet and outlet pressures so density dependence can be included). 
      
      .. math::
          \Delta P_{acc} = G^2\left\{\left[\frac{(1-x_o)^2}{\rho_{l,o}
          (1-\alpha_o)} + \frac{x_o^2}{\rho_{g,o}\alpha_o} \right]
          - \left[\frac{(1-x_i)^2}{\rho_{l,i}(1-\alpha_i)} 
          + \frac{x_i^2}{\rho_{g,i}\alpha_i} \right]\right\}
  
      Parameters
      ----------
      m : float
          Mass flow rate of fluid, [kg/s]
      D : float
          Diameter of pipe, [m]
      xi : float
          Quality of fluid at inlet, [-]
      xo : float
          Quality of fluid at outlet, [-]
      alpha_i : float
          Void fraction at inlet (area of gas / total area of channel), [-]
      alpha_o : float
          Void fraction at outlet (area of gas / total area of channel), [-]
      rho_li : float
          Liquid phase density at inlet, [kg/m^3]
      rho_gi : float
          Gas phase density at inlet, [kg/m^3]
      rho_lo : float, optional
          Liquid phase density at outlet, [kg/m^3]
      rho_go : float, optional
          Gas phase density at outlet, [kg/m^3]
  
      Returns
      -------
      dP : float
          Acceleration component of pressure drop for two-phase flow, [Pa]
          
      Notes
      -----
      The use of different gas and liquid phase densities at the inlet and outlet
      is optional; the outlet densities conditions will be assumed to be those of
      the inlet if they are not specified.
      
      There is a continuous variant of this method which can be integrated over,
      at the expense of a speed. The differential form of this is as follows 
      ([1]_, [3]_):
          
      .. math::
          - \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{
          (1-x)^2}{\rho_l(1-\alpha)} + \frac{x^2}{\rho_g\alpha}\right]
  
      Examples
      --------
      >>> two_phase_dP_acceleration(m=1, D=0.1, xi=0.372, xo=0.557, rho_li=827.1,
      ... rho_gi=3.919, alpha_i=0.992, alpha_o=0.996)
      706.8560377214725
      
      References
      ----------
      .. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
         Transfer, 3E. New York: McGraw-Hill, 1998.
      .. [2] Awad, M. M., and Y. S. Muzychka. "Effective Property Models for 
         Homogeneous Two-Phase Flows." Experimental Thermal and Fluid Science 33,
         no. 1 (October 1, 2008): 106-13. 
         doi:10.1016/j.expthermflusci.2008.07.006.
      .. [3] Kim, Sung-Min, and Issam Mudawar. "Review of Databases and
         Predictive Methods for Pressure Drop in Adiabatic, Condensing and
         Boiling Mini/Micro-Channel Flows." International Journal of Heat and
         Mass Transfer 77 (October 2014): 74-97.
         doi:10.1016/j.ijheatmasstransfer.2014.04.035.
      '''
      G = 4*m/(pi*D*D)
      if rho_lo is None:
          rho_lo = rho_li
      if rho_go is None:
          rho_go = rho_gi
      in_term = (1.-xi)**2/(rho_li*(1.-alpha_i)) + xi*xi/(rho_gi*alpha_i)
      out_term = (1.-xo)**2/(rho_lo*(1.-alpha_o)) + xo*xo/(rho_go*alpha_o)
      return G*G*(out_term - in_term)